# nt.number theory – Simultaneous reductions of elliptic curves: same number of points \$|E(Bbb F_p)| = |E'(Bbb F_p)|\$ for some prime \$p\$?

$$newcommand{End}{mathrm{End}} newcommand{Gal}{mathrm{Gal}} newcommand{kb}{overline{k}} newcommand{Z}{mathbb{Z}} newcommand{F}{mathbb{F}} newcommand{Q}{mathbb{Q}}$$

Let $$E,E’$$ be two elliptic curves over $$Q$$. Is there at least one prime number $$p geq 5$$ of good reduction for $$E,E’$$ such that $$|E(F_p)| = |E'(F_p)|$$ ?

If $$E,E’$$ are isogenous over $$Q$$, this is clearly true. In any case, François Charles proved that there are infinitely many primes $$p$$ such that the reductions of $$E,E’$$ mod $$p$$ are isogenous over $$overline{F_p}$$. If the result holds over $$F_p$$, we are done.

For instance, for the curves $$y^2 = x^{3} – 3 x – 3, y^2 = x^{3} + x + 6$$, the smallest such prime is $$p = 3121$$.

One could ask if the above question can be generalized to arbitrary smooth projective irreducible curves $$C,C’$$ (not necessarily of the same genus!) over $$Q$$, or even over a number field $$K$$ (though the heuristic only works well if the primes $$mathfrak{p}$$ are totally split over $$Q$$, or at least have degree $$f leq 2$$).

Using ideas from there, one can show that if $$E,E’$$ are not isogenous over $$overline{Q}$$, then the set of such primes has density $$0$$. According to Lang–Trotter heuristics, one should expect the number of such primes $$p to be $$sim c sqrt{x} / log(x)$$ for some $$c geq 0$$ (if $$c neq 0$$, this should give infinitely many such primes).

Note: in his talk, he mentions that he doesn’t know what happens for three elliptic curves, nor what happens over $$F_p$$ instead of $$overline{F_p}$$.
For four (or more) elliptic curves, one should not expect infinitely many primes such that $$|E_i(F_p)|$$ are all equal when $$1 leq i leq 4$$.

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